# The existence of $T$-numbers in positive characteristic

## TL;DR

This paper proves the existence of $T$-numbers in positive characteristic Laurent series, filling a key gap in the classification analogous to Mahler's for real numbers.

## Contribution

It establishes the existence of $T$-numbers in the classification of Laurent series over finite fields, which was previously an open problem.

## Key findings

- Confirmed the existence of $T$-numbers in positive characteristic.
- Extended Mahler's classification to Laurent series over finite fields.
- Provided a foundation for further study of Diophantine approximation in function fields.

## Abstract

As an analogue of Mahler's classification for real numbers, Bundschuh introduced a classification for Laurent series over a finite field, divided into $A,S,T,U$-numbers.It is known that each of $A,S,U$-numbers is nonempty.On the other hand, the existence of $T$-numbers is open.In this paper, we give an affirmative answer to the problem.